Requisites

Additional Requirements

Some acquaintance with algebraic structures (groups and fields – these will be used as examples) and with predicate languages and their structures, such as may be obtained from the above courses, will be assumed (though there will be a brief recap at the beginning of the course unit). It would be helpful, though not necessary, to have seen some basic model theory (such as in MATH3301).

Students are not permitted to take, for credit, MATH43051 in an undergraduate programme and then MATH63051 in a postgraduate programme at the University of Manchester, as the courses are identical.

Aims

To present the basic notions and results of model theory.

To illustrate these in a variety of types of examples.

To show some applications of the ultraproduct construction.

To show some applications of realising types in elementary extensions.

Overview

Model theory deals with those properties of mathematical structures which can be expressed using formulae of a formal predicate language. One theme is the investigation of the class of those structures which are the models of a set of sentences from predicate logic. Another theme is the analysis of definability in individual structures and the use of elementary extensions to produce non-standard elements. An example of the latter is producing infinitesimals in extensions of the set of real numbers, an infinitesimal being an element x satisfying x>0 and x<1/n for every positive integer n. The Compactness Theorem says that, since this (infinite) set of conditions is finitely satisfied in the field of real numbers, there is an elementary (=nice) extension of the reals which contains such an element.

We will introduce and use the ultraproduct construction, which is a way of producing, from a family of structures, an “average” structure (and it can be used to give a neat proof of the Compactness Theorem). Back-and-forth is an inductive method for building up maps between structures. We’ll use it to investigate the Random Graph (a countably infinite graph which contains a copy of every countable graph).

Types are descriptions, using the formal language, of elements and potential elements. We will see how these can be used to try to classify the models of a theory. A particularly nice case is when a theory is countably categorical, meaning that it has just one countable model up to isomorphism; we will characterise these theories in various ways (one being that the automorphism group of a countable model has only finitely many orbits on n-tuples). Compactness of the space of types is a key ingredient in the proof of these characterisations.

Learning outcomes

On successful completion of this course unit students will be able to

understand the relation between structure and syntax, definable sets and types, in the context of algebraic and relational structures;

understand the ultraproduct construction and how to use it;

be able to analyse examples from a model-theoretic perspective;

understand the fundamental results on the class of models of a theory in predicate logic;

Recommended reading

I will provide full course notes but there are quite a few texts on model theory around. For example, those below, but they are aimed at graduate students so don’t expect to move quickly when reading them. There are also sets of lecture notes on the web. So you can browse around and see what you like/what’s helpful.

Feedback methods

Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding. Coursework also provides an opportunity for students to receive feedback. Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.